Non-Probabilistic Structural Reliability Analysis Integrating the PSO and the Kriging Model
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摘要:
针对复杂结构可靠性分析中面临的隐式功能函数和小样本问题，提出了一种粒子群优化和Kriging模型相结合的结构非概率可靠性分析方法。采用多维椭球描述结构不确定参数，运用粒子群优化对模型相关参数进行求解，并构建隐式功能函数的Kriging模型进行可靠性分析。三个算例结果表明所提方法有效可行，精度和效率均优于基于Kriging模型的非概率可靠性分析方法。
Abstract:In view of the problems with implicit performance functions and limited experimental data in the reliability analysis of complex structures, a non-probabilistic reliability method combining the particle swarm optimization (PSO) with the Kriging model was presented. A multidimensional ellipsoid was first used to characterize the uncertain parameters of structures. The Kriging model was then constructed for the implicit performance function, wherein its optimal related parameters was determined through the PSO. Based on the proposed model the reliability analysis was explicitly conducted. The results of 3 numerical examples show that, the proposed method is of effectiveness and feasibility, and has higher accuracy and efficiency than those based on the traditional Kriging model.
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表 1 相关系数为0.2的迭代过程
Table 1. The iterative process for a correlation coefficient of 0.2
iteration step design point reliability index 0 (−2.4442, 4.2332) 4.8882 1 (−1.8607, 4.1064) 4.5083 2 (−1.8641, 4.1039) 4.5074 reference value (−1.8692, 4.1017) 4.5075 
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表 2 不同相关系数的可靠性指标（算例1）
Table 2. Reliability indexes for different correlation coefficients (example 1)
correlation coefficient PSO-Kriging iterations CPU time Kriging iterations CPU time reference value 0 4.1704 2 0.45703 s 4.1653 5 1.00112 s 4.1701 0.2 4.5073 2 0.38524 s 4.5002 6 1.20133 s 4.5075 0.5 5.2134 3 0.68554 s 5.2039 5 1.01367 s 5.2138 0.7 5.9237 3 0.88563 s 5.9355 7 1.40156 s 5.9235 0.9 7.0337 2 0.57889 s 7.0302 6 1.16778 s 7.0339
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表 3 相关系数为0.9的迭代过程
Table 3. The iterative process for a correlation coefficient of 0.9
iteration step design point reliability index 0 (0.0662, 0.0265, 0.0088, 0.00650.4059) 0.4123 1 (−0.0031, −0.0004, 0.0146, 0.0141, 0.3458) 0.3464 2 (−0.0040, −0.0005, 0.0124, 0.0176, 0.3415) 0.3422 3 (−0.0044, −0.0007, 0.00132, 0.0179, 0.3441) 0.3448 4 (−0.0042, −0.0007, 0.0134, 0.0181, 0.3439) 0.3447 reference value (−0.0058, −0.0015, 0.0146, 0.0155, 0.3437) 0.3444
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表 4 不同相关系数的可靠性指标（算例2）
Table 4. Reliability indexes for different correlation coefficients (example 2)
correlation coefficient PSO-Kriging iterations CPU time Kriging iterations CPU time reference value 0 0.3469 4 16.15572 s 0.3406 7 27.16953 s 0.3465 0.3 0.3504 5 20.19465 s 0.3466 9 34.93226 s 0.3509 0.5 0.3517 4 17.23898 s 0.3612 7 26.35562 s 0.3513 0.7 0.3487 5 21.56773 s 0.3454 8 31.05089 s 0.3490 0.9 0.3447 4 18.06577 s 0.3404 7 28.09947 s 0.3444
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表 5 MCS求解的95%置信区间（算例2）
Table 5. The 95% confidence intervals of MCS solutions (example 2)
MCS value mean value deviation lower bound upper bound 0.3465 0.3469 0.0029 0.3463 0.3475 0.3509 0.3506 0.0029 0.3500 0.3511 0.3513 0.3511 0.0030 0.3505 0.3517 0.3490 0.3491 0.0030 0.3485 0.3497 0.3444 0.3448 0.0031 0.3442 0.3454
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表 6 不确定变量的分布参数
Table 6. The distribution parameters of uncertain variables
uncertain variable mean value radius $ {A_1} $/mm2 700 70 $ {A_2} $/mm2 6200 620 $ {A_3} $/mm2 5300 530 $ {A_4} $/mm2 8800 880 $ L $/mm 15000 1500
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表 7 相关系数为0的迭代过程
Table 7. The iterative process for a correlation coefficient of 0
iteration step design point reliability index 0 (−0.0083, −0.4266, −0.3646, −0.0519, 0.8256) 0.9997 1 (−0.0109, −0.5652, −0.4831, −0.0587, 0.9338) 1.1952 2 (−0.0100, −0.5201, −0.4446, −0.0540, 0.8593) 1.0997 3 (−0.0089, −0.4827, −0.4126, −0.0478, 0.7875) 1.0870 4 (−0.0084, −0.5107, −0.4222, −0.0509, 0.8504) 1.0793 5 (−0.0105, −0.5065, −0.4289, −0.0465, 0.8372) 1.0694 6 (−0.0094, 0.5032, −0.4301, −0.0431, 0.8356) 1.0669 reference value (−0.0092, −0.5043, −0.4280, −0.0484, 0.8340) 1.0656
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表 8 不同相关系数的可靠性指标（算例3）
Table 8. Reliability indexes for different correlation coefficients (example 3)
correlation coefficient PSO-Kriging iterations CPU time Kriging iterations CPU time reference value 0 1.0669 6 27.02193 s 0.9944 9 37.73403 s 1.0656 0.3 1.2683 7 31.52558 s 1.1793 10 41.92578 s 1.2679 0.5 1.4906 6 28.02193 s 1.3997 10 40.44356 s 1.4917 0.7 1.9043 6 26.02193 s 1.8253 9 38.37882 s 1.9023 0.9 3.1439 7 31.52558 s 2.9934 11 46.11937 s 3.1431
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表 9 MCS求解的95%置信区间（算例3）
Table 9. The 95% confidence intervals of MCS solutions (example 3)
MCS value mean value deviation lower bound upper bound 1.0656 1.0659 0.0029 1.0653 1.0665 1.2679 1.2672 0.0028 1.2667 1.2679 1.4917 1.4919 0.0025 1.4914 1.4924 1.9023 1.9023 0.0028 1.9017 1.9029 3.1431 3.1430 0.0029 3.1424 3.1435
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